# Solved – Vector ARMA, trend and mean reversion

This question is addressed to expert in econometrics. I generally fit econometric models and statistical learning models to financial time series and some discretionary traders usually asked me if I try to "catch momentum or mean-reversion" and I always reply to them "both". Am I right?

Using the best fit, for instance using AIC to select a vector ARMA, I should be able to capture the dynamic of the financial time series both auto correlation and causality/cross-correlation. Depending on the input I could have a forecast telling me the prices are trending or mean-reverting.

I realised though that the vector ARMA can rarely catch both effect at the same time. For instance an AR(1) the simplest model is always a mean-reverting model (if stationary). An integrated AR(1) (ARIMA(1,1,0)) is always a momentum model, as the change of prices will converge to the mean change of price (if the mean is positive and the mean is large compare to the standard deviation, this is a momentum model). Can more complex models Vector ARMA achieve to catch both dynamic? Or is it always one or the other?

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It's one or the other, depending on how the parameter estimates come out. Here's an example. Say we're modelling log-returns. If we have an ARIMA(1,1,0) model: $$x_t = mu + phi(x_{t-1} – mu) + epsilon_t,$$ then the conditional mean/1-step ahead prediction is $$mu(1 – phi) + phi x_{t}$$. Let's always assume that $$mu > 0$$, and that we don't have to worry about trading costs.
1. If we assume that $$0 < phi < 1$$, then we are supposed to sell when $$x_t < frac{mu(phi-1)}{phi} < 0.$$ Notice that the right hand side is negative, so we are supposed to sell when we experienced a significantly negative return. On the other hand, we buy when that inequality is reversed. So we buy when it's positive, or not-so-negative.
2. Now if $$-1 < phi < 0$$ then we buy when $$x_t < frac{mu(phi-1)}{phi}.$$ The right hand side here is positive this time. This means we only don't buy when we experienced a significantly positive return.